Environmental Engineering Reference
In-Depth Information
The continuity term was later modifi ed by Rabotnov and was called the
damage parameter
ω
, where
⎛
⎜
⎞
A
A
[3.55 ]
1
⎛
⎝
ω
=−
.
0
By assuming a power-law dependence of stress, the creep rate at constant
temperature was described as
m
k
σ
0
[3.56 ]
ε
=
c
,
(
1
)
p
ω
−
where
m
and
p
are material parameters. At time
t
= 0,
ω
= 0 and the above
equation assumes the power-law form. As
increases, the creep rate
increases and when it achieves a critical value, the creep rate tends towards
infi nity and failure follows.
In order to describe the evolution of damage, Kachanov assumed that dam-
age is a function of the initial stress
ω
0
. This was later generalized by Rabotnov
who assumed that the damage is instead a function of the instantaneous stress
and described the rate of change of damage through the following:
σ
k
d
d
ω
B
σ
0
[3.57 ]
=
.
(
1
)
r
t
ω
−
Solving the above two equations gives the creep strain in the following
form:
⎡
⎤
1
⎛
⎝
⎞
⎟
λ
t
ε
⎡
⎤
⎢
⎥
[3.58 ]
c
11
−
⎛
⎜
⎛
⎝
=
−
⎢
⎢
⎥
ε
t
R
R
R
⎦
⎦
⎣
where
R
is the rupture strain,
t
is the
time and
t
R
is the time to rupture. The shape of the creep curve described by
Equation [3.58] is as shown in Fig. 3.23.
The damage tolerance parameter
ε
c
is the instantaneous creep strain,
ε
λ
is given by the following equation:
1
+
r
λ
=
[3.59 ]
.
1
+−
rp
−
The material fails in the steady-state creep regime when
= 1. Ashby and
Dyson
97
have demonstrated that each damage micromechanism has a char-
acteristic
λ
λ
and a characteristic shape of the creep curve. This implies that
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